On the projections of Ahlfors regular sets in the plane

This paper contains the following $\delta$-discretised projection theorem for Ahlfors regular sets in the plane. For all $C,\epsilon >0$ and $s\in [0,1]$, there exists $\kappa >0$ such that the following holds for all $\delta >0$ small enough. Let $\nu$ be a Borel probability measure on $S^1$ satisfying $\nu (B(x,r))\le C{r}^{\epsilon }$ for all $x\in S^1$ and $r>0$. Let $K\subset B(1)\subset {\mathbb{R}}^2$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $\theta \in \mathrm{s}\mathrm{p}\mathrm{t}\nu$ such that $$\mathrm{\mid }{\pi }_{\theta }(F){\mathrm{\mid }}_{\delta }\ge {\delta }^{\epsilon -s}$$ for all $F\subset K$ with $\mathrm{\mid }F{\mathrm{\mid }}_{\delta }\ge {\delta }^{\kappa -s}$. Here ${\pi }_{\theta }(z)=\theta \cdot z$ for $z\in {\mathbb{R}}^2$.